If you believed everything you
read about Godelís incompleteness theorem, you could be forgiven for
thinking it holds the key to life, the universe and everything. By some
accounts, this equation explains the human
mind, it determines the nature of free will and it limits what
we can ultimately know about the universe.

But there is really no need for
hype. Mathematicians generally agree that Godelís incompleteness theorem
is one of the most important achievements in mathematics. It is also one
of the most mysterious and disturbing.

At the end of the 19th
century mathematicians had huge optimism that a new kind mathematics was
emerging that would eventually explain everything the Universe could throw
at it. In 1920, the German mathematician David Hilbert proposed a research
program that would aim to build all of mathematics on solid and complete
mathematical foundations. He believed this process would show that
mathematics contained no contradictions, that in mathematical language, it
was consistent.

Hilbert was wrong. And the man who
proved it was Kurt Godel.

Godel discovered a property of any logical
system that truly astounded mathematicians. He began by thinking about the
way rules can be used to make statements. What Godel found
was that if these rules contain no contradictions then there is something
strange about the statements that can be made with them: certain
statements cannot be proved true using the available rules, even though
they are true. Instead extra rules are needed to prove the point. So the
original set of rules must be incomplete.

Godelís theorem is that if a set
of rules are consistent, they are incomplete. For example, arithmetic is a
set of rules for making statements about numbers. These rules contain no
contradictions and so by Godel's theorem must be incomplete.

At a stroke, the infallibility of
mathematics was shattered. The theorem means that it is not possible to
construct solid and complete foundations for mathematics in a way that
allows all mathematical truths to be proved.

The philosophical implications of
this are hotly debated. Many people have wondered whether the laws of
physics are a consistent set of rules. If so, Godelís theorem would apply
meaning they must be incomplete.
But little headway has been made in the process of formulating the laws of
physics in a consistent way and there is little agreement on whether it is
even possible. Could there exist laws of physics that are true and yet unprovable? Maybe.

Others have wondered what Godelís
theorem means for our understanding of the human mind. If our brains are
machines that work in a consistent way, then Godelís theorem applies. Does
that mean that it is possible to think of ideas that are true but be
unable to prove them? Nobody knows.